# Classical U(1) Lattice Gauge Theory in D=2

## TL;DR

This paper proves that in a 2D U(1) lattice gauge theory, configurations with no topological structure correspond to pure gauge fields with zero local curvature, with topological information captured by chart transitions and Chern numbers.

## Contribution

It establishes a rigorous connection between lattice configurations and classical gauge fields, highlighting the role of topology and bundle reconstruction.

## Key findings

- Any U(1) lattice configuration corresponds to a pure gauge field with zero local curvature.
- Topological information is encoded in chart transitions and Chern numbers.
- The Chern number depends on the bundle reconstruction and is uniquely defined under certain conditions.

## Abstract

Under the hypothesis of no topological structure below a certain scale, we prove that any U(1) lattice configuration corresponds to a classical U(1) gauge field with zero local field strength; i.e. any local representative of the pullback connection one-form is a pure gauge and the local curvature two-form is thus identical zero. The topological information is completely carried by the chart transitions. To each such U(1) lattice configuration we assign a Chern number, which generally depends on the reconstruction of the bundle and is only unique under certain restrictions.